Let X and Y be independent random variables, with the same probability distribution. Find P(X=Y) and P(X $\leq$ Y)

Question

By an exemple, try flipping two coin n times. X is the number of heads of the fist coin, Y is the number of tails of the second. We have, P(X=k)=$\binom{n}{k} \frac{1}{2^n}$=P(Y=k). Or if Y and X is the number of heads of diferent coins, we have trivially that P(X=k)=P(Y=k).

Answer 1

$P(X=Y)=P(X-Y=0)$ which is zero if you have continous distributions but in this case that's not what you're dealing with. You have to use discrete convolution.

The same applies for the other question, you have to use $P(X-Y<0)$ instead of $P(X<Y)$.

Answer 2

Since $X,Y$ are iid, the couple $(X,Y)$ has law : $P(X=p,Y=q) = P(X=p)P(Y=q) = \binom{n}{p} \binom{n}{q} \frac{1}{4^n}$.

Then, $P(X-Y=0) = \sum_{k=0}^n P(X-Y=0,Y=k)=\sum_{k=0}^n P(X=k,Y=k) = \sum_{k=0}^n \binom{n}{k}^2 \frac{1}{4^n}. $